Week10-2up - DISCUSSION SECTION Tuesday 11/5 EXAM 2 –...

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Unformatted text preview: DISCUSSION SECTION Tuesday 11/5: EXAM 2 – During your Monday 11/4: OPTIONAL review 8.61, 8.67–69 Thursday : Exercises 8.29, 8.33, 8.34, 8.38–41, 8.59, Wednesday : P. 334 – 338, 347–351, For Tomorrow : Exercises 8.18, 8.21–23, 8.25, 8.27 Today : P. 328 – 332 Assignments to within “B” units with confidence. otherwise use . (p. 309) , . (p. 322) 225 HYPOTHESIS, – (p. 322) , light bulb ex. The “other” hypothesis is called the NULL manner What we are “trying to prove” in an objective, fair light bulb ex. ALTERNATIVE or RESEARCH hypothesis, – The hypothesis of MAIN INTEREST is the Parts of a statistical test. (p. 322) . if you have one, and solve for Use “ballpark” value for ©  ! ! ¢ 3% Estimate    ( )  £1 and decide that nothing can be done. $2 individually, can do nothing, but collectively can meet, ¡ Last Time: ¦ 3 4% Thought: A committee is a group of people who, STA 2023 c D.Wackerly - Lecture 17   & '% ¢ &% ( ) 0 £1 224 $2 STA 2023 c D.Wackerly - Lecture 17  ¡ ¡ § ¦ ! £¢ $# " £ ¥ £¦ ¤ ¨ § ! Type I error Reject Ho Type I error Type II error (p. 325) when true In our lightbulb example, saying when ’s that we pick. The test that we will discuss have the SMALLEST for the 226 saying what we “want” to say when we should not accepting and/or Correct Type II error Ha true (p. 323), SIGNIFICANCE Correct Accept Ho  ¥ Reality Ho true LEVEL of the test. ¥ ! ¥ ! ! ¡  £ ¤¢ ¡ ¦ ! Decision ) 0 £1 ! Errors: p. 325 )  £1 ¥  ¢  ¥ &% ¥ ¦  ¦ ¦ £¥ ¢ ¦ ¥ ¢ 3 4% ¡ $2 ¥ ’s . . forms the basis for our decision. gives values of TS for which 4. Rejection Region : (RR) depends on the choice of Compute value of TS Get data Do experiment Then 227 Make decision is REJECTED computed from the sample data using a formula 3. Test Statistic : (TS) 2. Alternative Hypothesis : 1. Null Hypothesis : Parts of a Statistical Test (p. 326) STA 2023 c D.Wackerly - Lecture 17 ! ! § $2 ! ! ! § ! ! &% ¨ ©¥ STA 2023 c D.Wackerly - Lecture 17 § ¡ 3% ! 3% and because we usually do . . 228 , what kind of error could we judgement Don’t want to accept usually that is really true , so we reserve depends on the value of the parameter in – What is the probability of a TYPE II error? make? – If we accept do so. not know the probability of making an error if we – We do not ACCEPT REGION, we DO NOT REJECT If the value of the TS is NOT in the REJECTION REGION, we 3% ! If the value of the TS is in the REJECTION 3% ! Decision : STA 2023 c D.Wackerly - Lecture 17 Guilty : Innocent : Prosecutor : Experimenter Courtroom Analogy STA 2023 c D.Wackerly - Lecture 17 ! ! ! 3 4% Proof “beyond a reasonable doubt” : small. Put burden of proof on Prosecutor : Experimenter ! 3 4% 3% &% &% &% 3% ! ¥ 229 Test Statistic, TS Rejection Region, RR ! Need In this case, Lightbulb Example : a fixed particular value of about a Population Mean, Large Sample Hypothesis Testing &% 34% ( ) ( ) 0  3) 3) . 230 STA 2023 c D.Wackerly - Lecture 17 has an APPROXIMATE Recall : Large Sample ¢ is true has a If is true, is a TEST STATISTIC if is a That is 3% 3)  3% &% ( ) 0 £1 $2 ! ! ! ! )  3 4% ( )  £1 ! ©  $2 £¤  3) ) ( ¢¦  ©  ¢¦ STA 2023 c D.Wackerly - Lecture 17 ) ¢ £ ¨¤ § ¤ £ ¨¤ § ) ! ¡ 1 ¤ ¥ ) ¤ 3) ¢¦ ¤   £ §¦ ¢¦ ¤ # ¢¦ ¢¦  ) ¥ ¤ #¤ 3) 231 distribution distribution #¤ 3) ¤ ¤ 3) £ ¨¤ § ¥ ¥ ¢ If Should Rejection region : If we are interested in 0 zα Rejection region : α type I error Want ! ! ! ¦ “something” in favor of than . by a “lot” of standard errors. is probably than The true value of is is POSITIVE and LARGE true value is. © ! ! ! ¢ ) &% ( ) 0 ) ) © 0 § 3) 3 4% ( & 4% ( ) 0 )  true mean lifelength of ALL BULBS Ex. : Lightbulb Example “Upper Tail test”, “One Tail Test” (p. 329) ! Data : ! ! © , whatever that § ) ¢ )  ¨2  ¡ level test, RR :  is close to the true value of ¢ FACT:  £1 © ( ¦ § ¥ &% ( ) STA 2023 c D.Wackerly - Lecture 17 § ) 34% ( )  £1 0 £1 §  ¥ £ $2 1 232 ¤ $ $2 1  1 ¨ 3) 3% &% ¥ ) ( ( ) 0  © ( ¥  3) 3) 0 ) ¥ ¢¦ ¤ ¢¦ #¤ ¤ #¤ © ) ¥  $1 § STA 2023 c D.Wackerly - Lecture 17  233 3% at the level confidence ). at the is true!! – NOTE: This DOES NOT mean that with ? confidence ). level of level” ( or claim that the mean lifelength of all bulbs is larger than “ – In terms of this problem – – Conclusion : Is – RR : If we wanted significance” ( or with at the , is AT THE “ Claim that the mean lifelength of all bulbs, £1 § 2 §£ ¨ § ¨ 0 ¨ 1$ ! ¥ In terms of this problem: &% 2 ! LEVEL!! ¥ ! ) 235 and α − Zα If we are interested in : ? one-second runs? Use 0 . “refute the claim” based on data for on average, at least 10 boards per second”. Evidence to printed circuit boards claims that “product can inspect, Ex. : #8.24, p. 333 Manufacturer inspection equip. for STA 2023 c D.Wackerly - Lecture 17 &% ! 3% ( in favor of 234 ¥ § § ¦ Conclusion : § ¥ ¨ §£ ¨ ¨ STA 2023 c D.Wackerly - Lecture 17 “Lower Tail test”, (p. 329) ¨2 3 4% % $2 £1 &% 3% ( ¡  ¨ 2£ $2 ¥ ) ) ( © $2 ( )  £1 £ § 1 §£ ¨ &% ( ¡  ¢ ¨2  3) 3) ¢¦ ¤ © ( § ) #¤  ¢¦ ¤ #¤ 3% ( ¥ ) ¥ 10 6 7 8 level test level of significance. In terms of this 9 10 8 RR : 7 11 10 9 10 11 6 9 12 12 9 10 8 9 10 enough evidence at the boards inspected per second is less than 10 .” level to indicate that the mean number of circuit application: “There at the 9 10 0 12 10 12 11 7 9 9 13 9 9 9 Must have actual data (not just and ). 1-second runs. Data : 48 actual numbers Ex. # 8.24 Number of solder joints inspected in 48 Median 9.000 Maximum 13.000 Mean 9.292 Minimum 0.000 N 48 Q1 9.000 TrMean 9.432 Q3 10.000 StDev 2.103 SE Mean 0.304 237 Variable JntsInsp N 48 Mean 9.292 StDev 2.103 Test of mu = 10.000 vs mu < 10.000 The assumed sigma = 2.10 Z-Test SE Mean 0.304 Z -2.33 P 0.0099 Stat Basic Statistics 1 Sample Z, select (double click) variable; Click radio button Test Mean; Type in null value for mean; Select Alternative; Type StDev in box labelled Sigma, Click OK Variable JntsInsp Variable JntsInsp Descriptive Statistics Minitab File Open Worksheet; find M8-24.mtw in MiniData (double click). Stat Basic Statistics Display Descriptive Statistics, select (double click) variable, Click OK £ 9 11 10 ! § 8 12 ¢ ¢ 9 Hypothesis test for a mean. Minitab? STA 2023 c D.Wackerly - Lecture 17 ¢ 11 236 § 7 ¢ ¢ 10 10  ¡ ! ! ! £ § ¨$  ¢ 9 $ ¨1 ¤ ¨§ ¡ 10 ¤ ¡ Data: £ Back to #8.24: §  § § STA 2023 c D.Wackerly - Lecture 17  ¢ ¢  ¥ ! § ¨2 ¦  ¨ £$ 1 § § © ¨2 ¨2 § § ¨ ¨ ¤ 11$ ¤ DISCUSSION SECTION Tuesday 11/5: EXAM 2 – During your Monday 11/4: OPTIONAL review 8.61, 8.67–69 Thursday : Exercises 8.29, 8.33, 8.34, 8.38–41, 8.59, Today: P. 334–338, 347–351 Assignments makes a pretty small package – (John Ruskin) Thought: When a man is wrapped up in himself, he 238 UNKNOWN population mean Last Time: Large Sample Hypothesis Testing about STA 2023 c D.Wackerly - Lecture 18 ! ! STA 2023 c D.Wackerly - Lecture 18 Test Statistic © ) ! 3% ( OR estimator RR standard error Hypothesized Value from NULL hypothesis Sheet 239 hypothesized value Estimator and Standard Error from Formula  ¢ ) )  ) &%  3)  3) 3) ¥¤£¡¢ ¡ ¥ ¡ § §§ ¤ ¥¦¥¡¡ #¤ 3) ¤ ¡ 0  ©   0 © © ¥©£¡¡ ¨ ¥¥¡¡ ¤ © )  ( -z α/2 0 ¡ ! &% 3% α/2 ¥ ¥ How? &% ? 3% ( ¢ &% ( ) ) © ( and z α/2 α/2 or level? “Two Tailed test” (p. 331) ¤ ©  ¢¦ ¤ #¤ ) ¥  3) ¡ 3% pH is NOT that of neutral water at the 240 water specimens from a recreational lake. Can we claim that the mean indicates acidity. Randomly select £ © (  #¤ ¤  3) © ( ¥ © at the the not .” ¤ 241 level to indicate that the mean pH reading is enough evidence at level of significance. In terms of this application: “There Reject RR : level test: Back to pH example: STA 2023 c D.Wackerly - Lecture 18  Ex. : pH of 7 is neutral, over 7 is alkaline, under 7 1 § ¢¦ ¥ ) 0 §£ ¨  ! ! ! ¢¦ STA 2023 c D.Wackerly - Lecture 18 ¨ §£  1 ¨ ¥ § §£ 3 4% ©   ¨1 §£ ¨ ¨§ ¨$ § § § ¥ ¨ § £ Do you think that ? Recall the lightbulb example (P. 335) The p-value or observed significance level be rejected in favor of could ( IF we DO for which in favor of CONFIDENCE in our . What is the SMALLEST value of SO). decision to reject – Provides – Smaller to reject is chosen BEFORE the test is performed Hypothesis Testing STA 2023 c D.Wackerly - Lecture 18 Agrees with two-tailed test!! the 99% 242 confidence interval — the value “ ” is ? . ¡ £ confidence interval for  Construct a © ! ! &% ! ¥ If the mean pH is NOT , what is it? ¨ ) ¨  ¢ ¢ § ) ¦ ¥ ¨1 ¢ £ § ¥ $ ( £ ¤ ¥ ¤ !  ¥ ! &% STA 2023 c D.Wackerly - Lecture 18 ©  & 4% 3% ( )  ¨£ 3 4% 3% ( )  £1 0 £1 2 ¥ $2 $2 § 3% 243 3% &% the one observed p-value = true. indicative that Probability of a z-value Larger z-values are p-value !  ! ¦ rejection region STA 2023 c D.Wackerly - Lecture 18 ) ( ¨ ¨ ¨ is 244 – – – REJECT . . . . . . p-value allows him/her to assess the “rareness” of the observed event. 245 on a person who might be interested in your conclusions, the Instead of “imposing” YOUR CHOICE of – – . CANNOT reject REJECT In our case, p-value = .0256 p-value p-value STA 2023 c D.Wackerly - Lecture 18 ! &% ( )  £1 0 £1  ¥ ¨ ¦ § § ¨ ¡ ¡ 2 $2 © © © § 0 ¨£ 0 0 © 1 $2 2 2 § §$ §£ ¢ © ¡ § 0 0 ¨ 11$ ¨ $ © 0 § § 3 4% 3 4% 3% ! ! 3 4% ¥ ¥ ¡ ¡ ¥ ¥ ¥ ¥ ¥ § ¦ ¨1 ¨2 ¨£ §$ ¨ §£ ¨ ¡ ¡ ¡ ¡ 3% 3% 3% ¥ §  for any Variable JntsInsp N 48 Mean 9.292 StDev 2.103 Test of mu = 10.000 vs mu < 10.000 The assumed sigma = 2.10 SE Mean 0.304 ¡ 3% Z-Test ¥ for any that is § ! See page 234 of notes: 3% ! ! ! p-value 3% § Z -2.33 that is ¨ ¥ § . Smaller z-values are more indicative that § &% ( ) ©   ¡ £ ( ) £ § ¨ ¤ 11$ the one observed ¡ P 0.0099 ¨ . is true. – ? value = EX. : Have done a two-tailed test: and DOUBLE IT. 3% &% Probability of a z-value claim that with . . Find the area in whichever “tail” the -value is in © p-value § ! ! ¡ ! ¥ § TWO - Tailed Test § &% § STA 2023 c D.Wackerly - Lecture 18 Ex. : #8.24, P. 333 § ¤ ¨2 ( ) £ ( ) £  ¨ 2£ ) £ 246 © ¥ STA 2023 c D.Wackerly - Lecture 18 § ¨2 247 ! (1) true proportion of Diet Coke drinkers who select Diet Pepsi in a blind taste test? indicate that a majority of the Diet Coke drinkers will the taste of Diet Pepsi. Is there sufficient evidence to Coke and Diet Pepsi. indicated that they preferred Coke drinkers were given unmarked cups of both Diet How??? (2) would select Diet Pepsi in a blind taste test. . the proportion of Diet Coke drinkers who Estimate for # of trials ; number of trials in the sample size in the sample # of GOAL : Test hypotheses about trials based on a “large” the proportion of batteries that fail before select Diet Pepsi in a blind taste test. guarantee expires. 249 (Section 8.5) Recall the BINOMIAL EXPERIMENT. particular attribute ! ¡ UNKNOWN but FIXED PROPORTION of items with a  ¡ ¡    Ex. : #8.68, p. 352 In a “Pepsi Challenge”, 100 Diet ¢ 2 Interested in a POPULATION that contains an ¢ &% ( 3 4% ( Large Sample Tests About  ! ¦  ¤¡ STA 2023 c D.Wackerly - Lecture 18 ¡ 248 £ ¡ ! ¡ ¢ ¢ ¢ ¡  ¢ ¢ ! ¡ ¢ ¢ STA 2023 c D.Wackerly - Lecture 18 ¨ £  3¡ ! Consider testing distribution. 3 4% ( versus a fixed particular value of has an approximate OR OR ¡ ¡ ) That is  ¡ distribution ¤  ¥¤¥¡£¡¥¡¥¡¥¡¥¡¥¡¢ ¡ has an ¡ 3¡ is “large”  © ¡ ¡ ¡ ¥¦£¡¥¡¥¡¥¡¥¡¥¡¥¡¡ 3¡ 0  ¡ &% "¡  ¤ £ ¡  ¡ ¤  3¡ STATISTIC if is the null hypothesis, TEST hypothesized value standard error is true Rejection Regions (RR): , OR OR has a STANDARD Hypothesized Value from NULL hypothesis Sheet Estimator and Standard Error from Formula estimator NORMAL distribution If &% STA 2023 c D.Wackerly - Lecture 18 £¦¥¡¥¡¥¡¥¡¥¡¥¡¥¡¥¡£¡¥¡¡ § § 3 4%  ¥¤¥¡¥¡¥¡¥¡¥¡¥¡¥¡¥¡£¡¥¡¢ ¡ ¡ ¡ ¡ §§ §§ 3¡ If 250 ! 3% ¡ ©  ¡ ¢ ¤ 3¡ 3" ¢ ¡ 0 3¡ ¡ ! ! ¡ ( ©  ¤ ¦ © § §§ §§ STA 2023 c D.Wackerly - Lecture 18 ¡  3¡ 3¡ © © © §  3¡ ¡ ¤ © 0 © 0 © ¤ ©  ©    3¡   or 251 RR ¥©¥¡¥¡¥¡¥¡¥¡¥¡¥¡¥¡£¡¥¡¡ ¨ £¥¡¥¡¥¡¥¡¥¡¥¡¥¡¥¡£¡¥¡¡ 2 indicated that they preferred true proportion of all voters who think health care reform is the leading priority Data : £ SAMPLE of all Diet Coke drinkers. Note: the the Pepsi Challenge are a (4) (3) 252 is “large” Assumptions : the 100 individuals participating in level test, RR : § ! ! ©  § ! ¡ ¥ § &% £ ¡ ! select Diet Pepsi in a blind taste test? indicate that a majority of the Diet Coke drinkers will the taste of Diet Pepsi. Is there sufficient evidence to Coke and Diet Pepsi. Coke drinkers were given unmarked cups of both Diet Ex. : #8.68, p. 352 In a “Pepsi Challenge”, 100 Diet STA 2023 c D.Wackerly - Lecture 18  ¨2 ¡ ( 34% ( ¡  reject AT THE level of significance” ( or with Coke drinkers will select Diet Pepsi in a blind taste value? value = test. 253 claim that there is sufficient evidence at in favor of confidence ) to indicate that the majority of Diet the “ In terms of this problem: LEVEL!! Conclusion : STA 2023 c D.Wackerly - Lecture 18 ! ! ¡ ¨  2 ¨ 02  § ¡ § 3 4% ¤ § ¨2 §   § ¥ ! &% 2 ¨2 ¤ ! ¨ ! Basic Statistics 1 Proportion Sample 1 X 56 N 100 Sample p 0.560000 Test of p = 0.5 vs p > 0.5 90% CI (0.462710, 0.657290) Test and Confidence Interval for One Proportion Z-Value 1.20 P-Value 0.115 Click Box “Use test and interval based on normal distribution”, OK, OK Click Options, Select Alternative, Type in Null Value 254 ! Number of trials, Number of Successes Click radio button “Summarized Data”, type in Stat Minitab? STA 2023 c D.Wackerly - Lecture 18 ¢ ! ! ¢ ...
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